Solve 14z^3/7z^4

$$\frac { 14 z ^ { 3 } } { 7 z ^ { 4 } }$$

Evaluate

$$\frac{2}{z}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(14z^{3}\right)^{1}\times \frac{1}{7z^{4}}$$

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

$$14^{1}\left(z^{3}\right)^{1}\times \frac{1}{7}\times \frac{1}{z^{4}}$$

Use the Commutative Property of Multiplication.

$$14^{1}\times \frac{1}{7}\left(z^{3}\right)^{1}\times \frac{1}{z^{4}}$$

To raise a power to another power, multiply the exponents.

$$14^{1}\times \frac{1}{7}z^{3}z^{4\left(-1\right)}$$

Multiply 4 times -1.

$$14^{1}\times \frac{1}{7}z^{3}z^{-4}$$

To multiply powers of the same base, add their exponents.

$$14^{1}\times \frac{1}{7}z^{3-4}$$

Add the exponents 3 and -4.

$$14^{1}\times \frac{1}{7}\times \frac{1}{z}$$

Raise 14 to the power 1.

$$14\times \frac{1}{7}\times \frac{1}{z}$$

Multiply 14 times \frac{1}{7}.

$$2\times \frac{1}{z}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{14^{1}z^{3}}{7^{1}z^{4}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{14^{1}z^{3-4}}{7^{1}}$$

Subtract 4 from 3.

$$\frac{14^{1}\times \frac{1}{z}}{7^{1}}$$

Divide 14 by 7.

$$2\times \frac{1}{z}$$

Differentiate w.r.t. z

$$-\frac{2}{z^{2}}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{14}{7}z^{3-4})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}z}(2\times \frac{1}{z})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$-2z^{-1-1}$$

Do the arithmetic.

$$-2z^{-2}$$